homework1_sol

homework1_sol - ECO 475 HW1 Solution Yu LIU 1 Golden Rule y...

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Unformatted text preview: ECO 475 HW1 Solution Yu LIU September 11, 2010 1 Golden Rule y t = F ( k t ) k t +1 = (1 − δ ) k t + σy t (1) (a) The law of motion of capital in steady state: k ∗ = (1 − δ ) k ∗ + σF ( k ∗ ) δk ∗ = σF ( k ∗ ) (2) Since consumption is the amount of output not saved: c ∗ = (1 − σ ) F ( k ∗ ) = F ( k ∗ ) − σF ( k ∗ ) = F ( k ∗ ) − δk ∗ (3) (b) Using that c ∗ = F ( k ∗ ) − δk ∗ , we have that dc ∗ /dk ∗ = F ′ ( k ∗ ) − δ . Using implicit function theorem for δk ∗ = σF ( k ∗ ) , we have dk ∗ /dσ = F ( k ∗ ) δ − σF ′ ( k ∗ ) . (Assume δ − σF ′ ( k ∗ ) ̸ = 0 , which will be shown later.) Combining both by the chain rule we have: dc ∗ dσ = [ F ′ ( k ∗ ) − δ ] F ( k ∗ ) δ − σF ′ ( k ∗ ) (4) We need assumptions to ensure: 1. the existence of k ∗ > 2. c ∗ is rst increasing then decreasing in σ . For the rst requirement, assumptions are: (Please refer to the appendix for the proof.) (I) F : R + → R + is continuously di erentiable, concave, lim k ↘ F ′ ( k ) > δ σ , and lim k →∞ F ′ ( k ) < δ σ (II) k > is given. For the second requirement, Claim : Under assumption I, dk ∗ /dσ = F ( k ∗ ) δ − σF ′ ( k ∗ ) > . Sketched Proof : Let g ( k ) = σ δ F ( k ) . First note that there exists < k 1 < k ∗ such that g ( k 1 ) > k 1 . (5) 1 Since g is continuously di erentiable and concave, g ′ ( k ∗ ) ≤ g ( k ∗ ) − g ( k 1 ) k ∗ − k 1 = k ∗ − k 1 − g ( k 1 ) + k 1 k ∗ − k 1 = 1 − g ( k 1 ) − k 1 k ∗ − k 1 < 1 . (6) Then, δ − σF ′ ( k ∗ ) > . The denominator δ − σF ′ ( k ∗ ) is always positive and k ∗ is increasing in σ . So, for the second requirement, we only need to consider dc ∗ /dk ∗ = F ′ ( k ∗ ) − δ part. Note that k ∗ and σ move in the same direction. For low values of σ and k ∗ , lim k ↘ F ′ ( k ) > δ σ implies that F ′ ( k ∗ ) > δ . Then, dc ∗ dσ > ....
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homework1_sol - ECO 475 HW1 Solution Yu LIU 1 Golden Rule y...

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